Electronic Theory of Phase Transitions in Ca, Sr, and Ba under Pressure

The transitions induced by temperature and pressure between the face-centered cubic (fcc) and the body-centered cubic (bcc) phases in the alkaline earth metals—Ca, Sr, and Ba—are analyzed by computing the differences in Gibbs free energy between the two phases in the framework of the nearly-free-electron and harmonic approximations. At the absolute zero of temperature and pressure, the observed fcc in Sr and bcc in Ba were found to have the lower internal energy; in Ca, however, identical analysis led to lower energy in bcc rather than the observed fcc. An fcc-bcc transition characterized by a change in the sign of the difference in free energies in Sr at 0°K was found at a critical pressure $P_c\sim 10$ kbar; at zero pressure, it was found at a critical temperature, $T_c\sim 150°$K. Both these results agree only qualitatively with the observed $P_c\sim 36$ kbar and $T_c\sim 830°$K. Ca and Ba, already bcc at absolute zero, showed no phase transition.